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Banach space representations and Iwasawa theory. (English) Zbl 1006.46053

Authors’ abstract: We develop a duality theory between the continuous representations of a compact \(p\)-adic Lie group \(G\) in Banach spaces over a given \(p\)-adic field \(K\) and certain compact modules over the completed group ring \(o_K[[G]].\) We then introduce a “finiteness” condition for Banach space representations called admissibility. It is shown that under this duality, admissibility corresponds to finite generation over the ring \(K[[G]] := K\otimes o_K[[G]].\) Since this latter ring is Noetherian, it follows that the admissible representations of \(G\) form an Abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the group \(G := \text{GL}_2(\mathbb{Z}_p)\).

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
22E35 Analysis on \(p\)-adic Lie groups
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[1] Arnautov, V. I.; Ursul, M. I., On the uniqueness of topologies for some constructions of rings and modules, Siberian Mathematical Journal, 36, 631-644 (1995) · Zbl 0852.16031 · doi:10.1007/BF02107321
[2] Ballister, P. N.; Howson, S., Note on Nakayama’s Lemma for Compact Λ-modules, Asian Journal of Mathematics, 1, 224-229 (1997) · Zbl 0904.16019
[3] Bourbaki, N., Commutative Algebra (1972), Paris: Hermann, Paris · Zbl 0279.13001
[4] Bourbaki, N., General Topology (1989), Berlin: Springer, Berlin
[5] Bourbaki, N., Groupes et algèbres de Lie (1972), Paris: Hermann, Paris · Zbl 0244.22007
[6] Bourbaki, N., Topological Vector Spaces (1987), Berlin: Springer, Berlin · Zbl 0622.46001
[7] Curtis, C.; Reiner, I., Representation Theory of Finite Groups and Associative Algebras (1962), New York-London: Wiley, New York-London · Zbl 0131.25601
[8] Demazure, M.; Grothendieck, A., Schémas en Groupes I (1970), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0207.51401
[9] Diarra, B., Sur quelques représentations p-adiques de ℤ_p, Indagationes Mathematicae, 41, 481-493 (1979) · Zbl 0421.22013
[10] M. Lazard,Groupes analytiques p-adiques, Publications Mathématiques de l’Institut des Hautes Études Scientifiques26 (1965).
[11] Monna, A. F., Analyse non-archimedienne (1970), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0203.11501
[12] Schikhof, W. H., A perfect duality between p-adic Banach spaces and compactoids, Indagationes Mathematicae, 6, 325-339 (1995) · Zbl 0837.46062 · doi:10.1016/0019-3577(95)93200-T
[13] Schneider, P., Nonarchimedean Functional Analysis (2001), Berlin: Springer, Berlin
[14] P. Schneider and J. Teitelbaum,Locally analytic distributions and p-adic representation theory, with applications to GL_2,Preprintreihe des SFB 478, Heft 86, Münster, 1999. · Zbl 1028.11071
[15] Schneider, P.; Teitelbaum, J., U(g)-finite locally analytic representations, Representation Theory, 5, 111-128 (2001) · Zbl 1028.17007 · doi:10.1090/S1088-4165-01-00109-1
[16] Serre, J.-P., Local Fields (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0423.12016
[17] van Tiel, J., Espaces localement K-convexes I-III, Indagationes Mathematicae, 27, 249-258 (1965) · Zbl 0133.06502
[18] Trusov, A. V., Representations of the groups GL(2, ℤ_p)and GL(2, ℚ_p)in spaces over non-archimedean fields, Moscow University Mathematics Bulletin, 36, 65-69 (1981) · Zbl 0479.22007
[19] Washington, L. C., Introduction to Cyclotomic Fields (1982), Berlin: Springer, Berlin · Zbl 0484.12001
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